Simple operations with rational numbers
Adding rational numbers
When doing any mathematical operation in Algebra, use your rules to find the sign of your answer first, then add,
subtract, multiply, divide or raise a number to a power.
Remember our earlier suggestion that you write down your rules and definitions to use for future reference.
Rules for Addition
-
Like signs: add and put that sign
-
Unlike signs: subtract and put the sign of the biggest number
| Examples |
Explanation |
| 4 + 3 = 7 |
Think like signs; add; answer positive |
-4 + (-3) = -7 |
Think like signs; add; answer negative |
4 + (-3) = 1 |
Think unlike signs; subtract; answer positive because biggest number is positive |
-4 + 3 = -1 |
Think unlike signs; subtract; answer negative because the biggest number is negative |
Subtracting Rational Numbers
When doing any mathematical operation in Algebra, use your rules to find the sign of your answer first then add,
subtract, multiply, divide or raise a number to a power.
Remember our earlier suggestion that you write down your rules and definitions to use for future reference.
To develop the rule for subtraction, we need to develop another property of numbers.
Additive Inverse Property:
For every a, a + (-a) = 0
Where -a can be read as
-
The additive inverse of a
-
The opposite of a
-
The negative of a
| Examples |
Explanation |
| 5 and -5 |
Because 5 + (-5) = 0 remember
unlike sign; subtract
where a = 5 and -a
= -5 |
-7 and 7 |
Because -7 + 7 = 0
where a = -7 and -a
= 7 |
As you can see, -a is not always negative. It has the opposite sign of a. So if a is positive, then -a is negative; but, if a is negative then -a is positive.
Rules for Subtraction
-
To subtract a number, add its additive inverse
-
Change the sign and add
We will use "change the sign and add" as our way of remembering the rule for subtraction.
Again, a minus sign in front of a grouping symbol tells us to change the sign.
| Examples |
Explanation |
| -3 - (-5) |
Change the -5 to a plus 5 then add
unlike signs; subtract and put the sign of the biggest number |
-3 + 5
2 |
If there are no grouping symbols in a problem, then you add.
| Examples
|
Explanation |
| -3 - 5 |
|
-8 |
Like sign; add and put that sign |
If there is a plus sign in front of a grouping symbol, then rewrite as is and add.
| Examples
|
Explanation |
| -3 + (-5) |
Remember: order of operation tells us to do
grouping symbols first. |
-3 - 5 |
Because there is a plus in front of ( ) rewrite as -5 |
-8 |
Like signs; add and put that sign |
The Commutative and Associative properties allow us to add numbers in any order we choose. If you are adding
more than two numbers, I suggest that you:
-
Add all the positive numbers
-
Add all the negative numbers
-
Subtract the two answers
Examples
| 1.
|
5 - 3 + 7 + 4 - 8 - 4
|
Explanation |
|
16 - 15 |
Rename 5 + 7+ 4 as 16
Rename -3 - 8 - 4 as -15 |
|
1 |
Unlike signs; subtract and put the sign of the biggest
number |
2.
|
-5 - (-3) - 7 - (-8)
|
|
|
-5 + 3 - 7 + 8 |
-( ) tells us to change the sign
-3 to 3, -8 to 8 |
|
11 - 12 |
Add 3 + 8 = 11, add -5 -7 = -12 |
|
-1 |
Subtract and put the sign of the biggest number |
Adding and subtracting like terms
We can also use our rules for addition and subtraction to combine like terms.
Remember that like terms are:
-
Same variable(s)
-
Same exponent(s) on those variable(s)
Examples: Simplify
| 1.
|
-3x + 5x
|
Explanation |
|
2x |
Unlike signs; subtract and put the sign of the biggest number |
2.
|
-a + 2a - 4a
|
Remember: -a = -1a |
|
-5a + 2a |
Add the -a - 4a = -5a |
|
-3a |
Subtract -5a + 2a = -3a |
Try these exercises
Add or subtract
-
-3 + 5
-
3 - 5
-
7 - (-6)
-
-8 - (-4)
-
8m + 6m
-
-43 + 16
-
3x - 8x + 7x - 10x
-
8y - (-8y)
-
8 + (-4) + 7
-
-6 - 9
Answers to questions
-
-3 + 5 = 2
-
3 - 5 = -2
-
7 - (-6)
7 + 6
13
-
-8 - (-4)
-8 + 4
-4
-
8m + 6m
14m
-
-43 + 16
-27
-
3x - 8x + 7x -10x
10x -18x
-8x
-
8y - (-8y)
8y + 8y
16y
-
8 + (-4) + 7
15 - 4
11
-
-6 - 9
-15